The Tiling of the Hyperbolic 4D Space by the 120-cell is Combinatoric

The splitting method was defined by the author in (Margenstern 2002a, Margenstern 2002d). It is at the basis of the notion of combinatoric tilings. As a con- sequence of this notion, there is a recurrence sequence which allows us to compute the number of tiles which are at a fixed distance from a given tile. A polynomial is attached to the sequence as well as a language which can be used for implementing cellular automata on the tiling. The goal of this paper is to prove that the tiling of hyperbolic 4D space is combinatoric. We give here the corresponding polynomial and, as the first consequence, the language of the splitting is not regular, as it is the case in the tiling of hyperbolic 3D space by rectangular dodecahedra which is also combinatoric. 1

[1]  Maurice Margenstern,et al.  On the Infinigons of the Hyperbolic Plane, A combinatorial approach , 2002, Fundam. Informaticae.

[2]  Maurice Margenstern,et al.  A universal cellular automaton in the hyperbolic plane , 2003, Theor. Comput. Sci..

[3]  A. F. Adams,et al.  The Survey , 2021, Dyslexia in Higher Education.

[4]  Aviezri S. Fraenkel,et al.  Systems of numeration , 1983, IEEE Symposium on Computer Arithmetic.

[5]  Maurice Margenstern.  Tiling the Hyperbolic Plane with a Single Pentagonal Tile , 2002, J. Univers. Comput. Sci..

[6]  Maurice Margenstern,et al.  Initialising Cellular Automata in the Hyperbolic Plane , 2004, IEICE Trans. Inf. Syst..

[7]  Maurice Margenstern,et al.  Time and Space Complexity Classes of Hyperbolic Cellular Automata , 2004, IEICE Trans. Inf. Syst..

[8]  Maurice Margenstern,et al.  Register Cellular Automata in the Hyperbolic Plane , 2004, Fundam. Informaticae.

[9]  David B. A. Epstein,et al.  Word processing in groups , 1992 .

[10]  D. M. Y. Sommerville,et al.  An Introduction to The Geometry of N Dimensions , 2022 .

[11]  Maurice Margenstern.  A Combinatorial Approach to Hyperbolic Geometry as a New Perspective for Computer Science and Technology , 2003, Computers and Their Applications.

[12]  Maurice Margenstern,et al.  Tools for devising cellular automata in the hyperbolic 3D space , 2003, Fundam. Informaticae.

[13]  Maurice Margenstern,et al.  A Polynomial Solution for 3-SAT in the Space of Cellular Automata in the Hyperbolic Plane , 1999, J. Univers. Comput. Sci..

[14]  Maurice Margenstern,et al.  New Tools for Cellular Automata in the Hyperbolic Plane , 2000, J. Univers. Comput. Sci..

[15]  Maurice Margenstern,et al.  Fibonacci Type Coding for the Regular Rectangular Tilings of the Hyperbolic Plane , 2003, J. Univers. Comput. Sci..

[16]  M. Hollander,et al.  Greedy Numeration Systems and Regularity , 1998, Theory of Computing Systems.

[17]  A. Phillips.  The macmillan company. , 1970, Analytical chemistry.

[18]  John Stillwell,et al.  The Story of the 120-Cell, Volume 48, Number 1 , 2000 .

[19]  D. Hilbert,et al.  Geometry and the Imagination , 1953 .

[20]  Maurice Margenstern,et al.  Cellular Automata and Combinatoric Tilings in Hyperbolic Spaces. A Survey , 2003, DMTCS.

[21]  吴德恒,et al.  经Co , 1964 .

[22]  Maurice Margenstern.  Can Hyperbolic Geometry Be of Help for P Systems? , 2003, Workshop on Membrane Computing.

[23]  Maurice Margenstern,et al.  NP problems are tractable in the space of cellular automata in the hyperbolic plane , 2001, Theor. Comput. Sci..

[24]  H. Piaggio.  An Introduction to the Geometry of N Dimensions , 1930, Nature.